The present invention relates to communications equipment, and more particularly to a blind adaptive filtering algorithm for receivers of communication systems without need of training sequences.
In wireless multi-user communication systems, antenna array based non-blind linear minimum mean square error (LMMSE) receivers utilize long training sequences for the suppression of multiple access interference (MAI) due to multiple users sharing the same channel and the suppression of intersymbol interference (ISI) due to channel distortion in wired communications and multi-path propagation in wireless communications. The MAI suppression and ISI suppression are crucial to the performance improvement of wireless communication systems. However the use of training sequences not only costs system resources but also involves synchronization issues and other cumbersome procedures. Therefore, receivers for blind mode (without need of training sequences) with good performance (of suppression of both MAI and ISI), and low complexity are of extreme importance in communication systems.
In digital communication systems, by sampling or matched filtering, the received continuous-time signal can be transformed to a discrete-time signal which can be approximated as the output signal of a discrete-time multi-input multi-output (MIMO) linear time-invariant (LTI) system on which a variety of detection and estimation algorithms are based. The proposed blind adaptive filtering algorithm for receivers of communication systems is also based on the discrete-time MIMO LTI system model.
Blind equalization (deconvolution) of an MIMO LTI system, denoted H[n] (P×K matrix) is a problem of estimating the vector input u[n]=(u1[n], u2[n], . . . , uK[n])T (K inputs), where the superscript ‘T’ denotes the transposition, with only a set of non-Gaussian vector output measurements x[n]=(x1[n], x2[n], xP[n])T (P outputs) as follows and as shown in FIG. 1:
                                                                        x                ⁡                                  [                  n                  ]                                            =                            ⁢                                                                    H                    ⁡                                          [                      n                      ]                                                        *                                      u                    ⁡                                          [                      n                      ]                                                                      +                                  w                  ⁡                                      [                    n                    ]                                                                                                                          =                            ⁢                                                                    ∑                                          k                      =                                              -                        ∞                                                              ∞                                    ⁢                                                            H                      ⁡                                              [                        k                        ]                                                              ⁢                                          u                      ⁡                                              [                                                  n                          -                          k                                                ]                                                                                            +                                  w                  ⁡                                      [                    n                    ]                                                                                                                          =                            ⁢                                                                    ∑                                          j                      =                      1                                        K                                    ⁢                                                                                    h                        j                                            ⁡                                              [                        n                        ]                                                              *                                                                  u                        j                                            ⁡                                              [                        n                        ]                                                                                            +                                  w                  ⁡                                      [                    n                    ]                                                                                                          (        1        )            where ‘*’ denotes the discrete-time convolution operator, hj[n] is the jth column of H[n], and w[n]=(w1[n],w2[n], . . . , wP[n])T (P×1 vector) is additive noise. For ease of later use, let us define the following notations                cum{y1,y2, . . . ,yP} pth-order cumulant of random variables y1,y2, . . . ,yP         cum{y:p, . . . }=cum{y1=y, y2=y, . . . , yP=y, . . . }        Cp,q{y}=cum{y:p, y*:q} (y* is complex conjugate of y)        υj=(υj[L1], υj[L1+1], . . . , υj[L2])T ((L=L2−L1+1)×1 vector)        ν=(υ1T, υ2T, . . . , υPT)T         xj[n]=(xj[n−L1], xj[n−L1−1], . . . , xj[n−L2])T         {tilde over (x)}[n]=(x1T[n], x2T[n], . . . , xPT[n])T         {tilde over (R)}=E[{tilde over (x)}*[n]{tilde over (x)}T[n]] (expected value of {tilde over (x)}*[n]{tilde over (x)}T[n])        
Assume that we are given a set of measurements x[n], n=0, 1, . . . , N−1 modeled by (1) with the following assumptions:    (A1) uj[n] is zero-mean, independent identically distributed (i.i d.) non-Gaussian with variance E[|uj[n]|2]=σu2 and (p+q)th-order cumulant Cp,q{uj[n]}≠0, and statistically independent of uk[n] for all k≠j, where p and q are nonnegative integers and p+q≧3.    (A2) The MIMO system H[n] is exponentially stable.    (A3) The noise w[n] is zero-mean Gaussian and statistically independent of u[n].
Let v[n]=(υ1[n], υ2[n], . . . , υP[n])T denote a P×1 linear finite impulse response (FIR) equalizer of length L=L2−L1+1 for which v[n]≠0 (P×1 zero vector) for n=L1, L1+1, . . . , L2. Then the output of e[n] of the FIR equalizer v[n] can be expressed as
                                                                        e                ⁡                                  [                  n                  ]                                            =                            ⁢                                                ∑                                      j                    =                    1                                    P                                ⁢                                                                            υ                      j                                        ⁡                                          [                      n                      ]                                                        *                                                            x                      j                                        ⁡                                          [                      n                      ]                                                                                                                                              =                            ⁢                                                ∑                                      j                    =                    1                                    P                                ⁢                                                      υ                    j                    T                                    ⁢                                                            x                      j                                        ⁡                                          [                      n                      ]                                                                                                                                              =                            ⁢                                                v                  T                                ⁢                                                      x                    ~                                    ⁡                                      [                    n                    ]                                                                                                          (        2        )            Tugnait (see “Identification and deconvolution of multichannel linear nonGaussian processes using higher-order statistics and inverse filter criteria,” IEEE Trans. Signal Processing, vol. 45, no. 3, pp. 658–672, March 1997) proposed inverse filter criteria (IFC) for blind deconvolution of MIMO systems using second- and third-order cumulants or second- and fourth-order cumulants of inverse filter (i.e., equalizer) output Chi and Chen (see “MIMO inverse filter criteria and blind maximum ratio combining using HOS for equalization of DS/CDMA systems in multi-path,” Proc. Third IEEE Workshop on Signal Processing Advances in Wireless Communications, Taoyuan, Taiwan, Mar. 20–23, 2001, pp. 114–117, see also “Cumulant-based inverse filter criteria for MIMO blind deconvolution: properties, algorithms, and application to DS/CDMA systems in multi-path,” IEEE Trans. Signal Processing, vol. 49, no. 7, pp. 1282–1299, July 2001, and “Blind beamforming and maximum ratio combining by kutosis maximization for source separation in multi-path,” Proc. Thrid IEEE Workshop on Signal Processing Advances in Wireless Communications, Taoyuan, Taiwan, March 20–23, 2001, pp. 243–246) find the optimum v by maximizing the following IFC
                                          J                          p              ,              q                                ⁡                      (            v            )                          =                                                                        C                                  p                  ,                  q                                            ⁢                              {                                  e                  ⁡                                      [                    n                    ]                                                  }                                                                                                                                    C                  1.1                                ⁢                                  {                                      e                    ⁡                                          [                      n                      ]                                                        }                                                                                                  (                                  p                  +                  q                                )                            /              2                                                          (        3        )            where p and q are nonnegative integers and p+q≧3, through using gradient type iterative optimization algorithms because all Jp,q(ν) are highly nonlinear functions of ν (without closed-form solutions for the optimum ν). Note that the IFC given by (3) include Tugnait's IFC for (p,q)=(2,1) and (p,q)=(2,2) as special cases. Under the following conditions:                (a1) signal-to-noise ratio (SNR) is infinity,        (a2) the length of equalizer is infinite,the optimum equalizer output turns out to be one of the K input signals except for an unknown scale factor and an unknown time delay (i.e., the optimum equalizer is a perfect equalizer for one of the K input signals).        
Yeung and Yau (see “A cumulant-based super-exponential algorithm for blind deconvolution of multi-input multi-output systems” Signal Processing, vol. 67, no. 2, pp 141–162, 1998) and Inouye and Tanebe (see “Super-exponential algorithms for mulichannel blind deconvolution,” IEEE Trans. Signal Processing, vol. 48, no.3, pp. 881–888, March 2000) also proposed an iterative super-exponential algorithm (SEA) for blind deconvolution of MIMO systems. The iterative SEA updates ν at the Ith iteration by
                              v          I                =                                                            σ                u                            ·                                                R                  ~                                                  -                  1                                                      ⁢                                          d                ~                                            I                -                1                                                                                                          d                  ~                                                  I                  -                  1                                H                            ⁢                                                R                  ~                                                  -                  1                                            ⁢                                                d                  ~                                                  I                  -                  1                                                                                        (        4        )            where ∥a∥ denotes the Euclidean norm of vector a and{tilde over (d)}I−1=cum{eI−1[n]:r,(eI−1[n])*:s−1, {tilde over (x)}*[n]}  (5)in which r and s−1 are nonnegative integers, r+s≧3 and eI−1[n] is the equalizer output obtained at the (I−1)th iteration. Again, under the conditions (a1) and (a2), the designed equalizer by the SEA is also a perfect equalizer (for one of the K input signals) with a super-exponential convergence rate. A remark regarding the distinctions of gradient type IFC algorithms and the SEA are as follows:    (R1) Although the computationally efficient SEA converges fast (with a super-exponential convergence rate) for SNR=∞ and sufficiently large N, it may diverge for finite N and finite SNR. Moreover, with larger computational load than updating the equalizer coefficients ν by (4) at each iteration, gradient type iterative IFC algorithms (such as Fletcher-Powell algorithm (see D. M. Burley, Studies in Optimization, Falsted Press, New York, 1974)) always converge slower than the iterative SEA for p+q=r+s as x[n] is real and for (p,q)=(r,s) as x[n] is complex.
Therefore a need exists for a fast IFC based algorithm with performance similar to that of SEA with guaranteed convergence for finite SNR and finite data length, and is therefore suitable for a variety of practical uses.